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Full Description
Oriented matroids appear throughout discrete geometry, with applications in algebra, topology, physics, and data analysis. This introduction to oriented matroids is intended for graduate students, scientists wanting to apply oriented matroids, and researchers in pure mathematics. The presentation is geometrically motivated and largely self-contained, and no knowledge of matroid theory is assumed. Beginning with geometric motivation grounded in linear algebra, the first chapters prove the major cryptomorphisms and the Topological Representation Theorem. From there the book uses basic topology to go directly from geometric intuition to rigorous discussion, avoiding the need for wider background knowledge. Topics include strong and weak maps, localizations and extensions, the Euclidean property and non-Euclidean properties, the Universality Theorem, convex polytopes, and triangulations. Themes that run throughout include the interplay between combinatorics, geometry, and topology, and the idea of oriented matroids as analogs to vector spaces over the real numbers and how this analogy plays out topologically.
Contents
1. Realizable oriented matroids; 2. Oriented matroids; 3. Elementary operations and properties; 4. The topological representation theorem; 5. Strong maps and weak maps; 6. Single-element extensions; 7. The universality theorem; 8. Oriented matroid polytopes; 9. Subdivisions and triangulations; 10. Spaces of oriented matroids; 11. Hints on selected exercises; References; Index.
